From: <wtait at ix.netcom.com>
>The `tired old objection' assumes that if the natural numbers are some
>one thing, then they must be sets or something else other than just the
>numbers; and then continues by pointing out that there is no
>distinguished such thing that they can be. Dedekind's answer is that they
>are simply the numbers. The proper answer to *what* they are is
>answered---as he answered it---by exhibiting the structure of the system
>of numbers.
Oh, that's a very nice bit of history I didn't know. So when arguing
that the real line is better understood in terms of its structure than
its elements (the standard category theory line, and the only view of the
reals that I've been able to make any sense out of), one can cite Dedekind
as an early source of that insight (and you as endorsing that viewpoint).
This is very satisfactory.
The fact that the complement of the cuts would work just as well as the
cuts themselves as representations of the reals should make it clear
that it is meaningless to identify the reals with the cuts.
As far as the natural numbers go, and more generally the ordinals,
I am not a complete zealot about distinguishing representation and
identification, and confess to some fondness for identifying an ordinal
with its von Neumann representation (the set of all smaller ordinals).
Monogamy is easier with von Neumann ordinals than Dedekind cuts.
Vaughan Pratt